While the oven preheats, melt the butter in a small saucepan, along with the balsamic vinegar, honey, and a hefty pinch of salt. Bring to a boil, then reduce to a simmer and cook about 5 minutes, or until slightly thickened.
Place the figs, cut side up, in a baking dish the size of a pie pan. Top each fig half with a 1/2 tsp to 1 tsp of goat cheese. Drizzle the balsamic vinegar syrup over the figs.
Roast in the oven until very soft, 10 to 15 minutes.
Arrange on a platter and sprinkle with flaky salt.
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